Warp Bridge Sampling: The Next Generation

Abstract

Bridge sampling is an effective Monte Carlo method for estimating the ratio of normalizing constants of two probability densities, a routine computational problem in statistics, physics, chemistry, and other fields. The Monte Carlo error of the bridge sampling estimator is determined by the amount of overlap between the two densities. In the case of uni-modal densities, Warp-I, II, and III transformations (Meng and Schilling, 2002) are effective for increasing the initial overlap, but they are less so for multi-modal densities. This paper introduces Warp-U transformations that aim to transform multi-modal densities into Uni-modal ones without altering their normalizing constants. The construction of a Warp-U transformation starts with a Normal (or other convenient) mixture distribution $\phi_{\text{mix}}$ that has reasonable overlap with the target density $p$, whose normalizing constant is unknown. The stochastic transformation that maps $\phi_{\text{mix}}$ back to its generating distribution $N(0,1)$ is then applied to $p$ yielding its Warp-U version, which we denote $\tilde{p}$. Typically, $\tilde{p}$ is uni-modal and has substantially increased overlap with $N(0,1)$. Furthermore, we prove that the overlap between $\tilde{p}$ and $N(0,1)$ is guaranteed to be no less than the overlap between $p$ and $\phi_{\text{mix}}$, in terms of any $f$-divergence. We propose a computationally efficient method to find an appropriate $\phi_{\text{mix}}$, and a simple but effective approach to remove the bias which results from estimating the normalizing constants and fitting $\phi_{\text{mix}}$ with the same data. We illustrate our findings using 10 and 50 dimensional highly irregular multi-modal densities, and demonstrate how Warp-U sampling can be used to improve the final estimation step of the Generalized Wang-Landau algorithm (Liang, 2005), a powerful sampling and estimation method.